Germano Resconi

Hierarchical uncertainty metatheory based upon modal logic

Abstract This paper is intended to contribute to the formal study of uncertainty from u broad perspective. Its aim is to demonstrate that, in addition to prepositional calculus and probability theory, both fuzzy set theory and Dempster-Shafer evidence theory can be represented by the formal and semantic structures of modal logic. In particular, it is shown that the concept of multiple worlds in modal logic can be employed for constructing membership-grade functions of fuzzy sets, as well as belief and plausibility measures of evidence theory. It is also shown that additional theories of uncertainty, which have not been considered us yet. emerge naturally from the framework of modal logic. When looking at these various uncertainty theories emerging from modal logic from a metatheoretical perspective, a hierarchical ordering of the theories is recognized. We refer to the hierarchically ordered collection of uncertainly theories captured within the realm of modal logic as hierarchical uncertainty metatheory....

ON THE INTEGRATION OF UNCERTAINTY THEORIES

Investigations pursued in this paper contribute to a research project introduced by Resconi, Klir, and St. Clair [1] whose purpose is to employ syntactic and semantic structures of modal logic as a unifying framework within which various uncertainty theories can be formalized, compared, and organized hierarchically. This paper focuses on the explicit use of modal logic semantics to formalize fuzzy sets, belief measures, plausibility measures, and Sugeno λ-measures.

ON THE COMPUTATION OF UNCERTAINTY MEASURE IN DEMPSTER-SHAFER THEORY

An algorithm for computing the recently proposed measure of uncertainly AU for Dempster-Shafer theory is presented. The correctness of the algorithm is proven. The algorithm is illustrated by simple examples. Some implementation issues are also discussed.

Interpretations of various uncertainty theories using models of modal logic: a summary

Abstract This paper summarizes our efforts to establish the usual semantics of propositional modal logic as a unifying framework for various uncertainty theories. Interpretations for fuzzy set theory, Dempster-Shafer theory, probability theory, and possibility theory are discussed. Some properties of these interpretations are also presented, as well as directions for future research.

On modal logic interpretation of Dempster–Shafer theory of evidence

This article further develops one branch of research initiated in an article by Resconi, Klir, and St. Clair (G. Resconi, G. J. Klir, and U. St. Clair, Int. J. Gen. Syst., 21(1), 23‐50 (1992) and continued in another article by Resconi et al. (Int. J. Uncertainty, Fuzziness and Knowledge‐Based Systems, 1(1), 1993). It fully formulates an interpretation of the Dempster‐Shafer theory in terms of the standard semantics of modal logic. It is shown how to represent the basic probability assignment function as well as the commonality function of the Dempster‐Shafer theory by modal logic and that this representation is complete for rational‐valued functions (basic assignment, belief, or plausibility functions).

A general systems logical theory

Abstract The General System Logical Theory (GSLT) is obtained by combining Resconis logical theory of systems with Jessels theory of secondary sources. In the present paper we give a first account of GSLT, of its foundation, its main features, and most obvious applications. GSLT is defined by its aims and concretized by a new specific concept, that of an Elementary Logical System (ELS). ELS may be connected with Lie algebras. The systems formerly dealt with by Resconis and Jessels separate theories are identified as particular ELS. Subsequently are built up various networks of ELS, leading thus to natural and powerful extensions of the classical feedback theory. Finally GSLT is applied to three very different topics: wave propagations (or any physical nature), Riemann geometries and chemical controls, showing thus its versatility and usefulness.

Interpreting and extracting fuzzy decision rules from fuzzy information systems and their inference

Information systems, which contain only crisp data, precise and unique attribute values for all objects, have been widely investigated. Due to the fact that in realworld applications imprecise data are abundant, uncertainty is inherent in real information systems. In this paper, information systems are called fuzzy information systems, and formalized by (objects; attributes; f), in which f is a fuzzy set and expresses some uncertainty between an object and its attribute values. To interpret and extract fuzzy decision rules from fuzzy information systems, the meta-theory based on modal logic proposed by Resconi et al. is modified. The modified meta-theory not only expresses uncertainty between objects and their attributes, but also uncertainty in the process of recognizing fuzzy information systems. In addition, according to perception computing (proposed by Zadeh), granules of fuzzy information systems can be represented by fuzzy decision rules, so that, fuzzy inference methods can be used to obtain the decision attribute of a new object. Finally, a novel way of combining evidences based on the modified meta-theory is introduced, which extends the concept of combining evidences based on Dempster-Shafer theory.

FIELD THEORY AND MODAL LOGIC BY SEMANTIC FIELDS TO MAKE UNCERTAINTY EMERGE FROM INFORMATION

The new notion of semantic fields for modal logic is introduced, by which a degree of significance can be assigned to any possible world. Available information is the main source of the significance that shows the logical structure of the information itself. Thus, a guide is obtained to use information and also to discover and measure the uncertainty located in it. The worlds are useful for separating such information into its important parts. Any sentence is evaluated in worlds that are located at different levels of significance. An accessibility relation exists between the more significant and the less significant world. In this way, possible-worlds models of information are made possible using the semantic field. Uncertainty is given, normally, without any logic but with set theory. Thus, the semantic field is used to connect modal logic with set theory in order to obtain a better description of uncertainty.

Granular Reasoning Using Zooming In & Out

The concept of granular computing is applied to propositional reasoning. Such kind of reasoning is called granular reasoning in this paper. For the purpose, two operations called zooming in & out is introduced to reconstruct granules of possible worlds.

Agents’ model of uncertainty

Multi-agent systems play an increasing role in sensor networks, software engineering, web design, e-commerce, robotics, and many others areas. Uncertainty is a fundamental property of these areas. Agent-based systems use probabilistic and other uncertainty models developed earlier without explicit consideration of agents. This paper explores the impact of agents on uncertainty models and theories. We compare two methods of introducing agents to uncertainty theories and propose a new theory called the agent-based uncertainty theory (AUT). We show advantages of AUT for advancing multi-agent systems and for solving an internal fundamental question of uncertainty theories, that is identifying coherent approaches to uncertainty. The advantages of AUT are that it provides a uniform agent-based representation and an operational empirical interpretation for several uncertainty theories such as rough set theory, fuzzy sets theory, evidence theory, and probability theory. We show also that the introduction of agents to intuitionist uncertainty formalisms can reduce their conceptual complexity. To build such uniformity the AUT exploits the fact that agents as independent entities can give conflicting evaluations of the same attribute. The AUT is based on complex aggregations of crisp (non-fuzzy) conflicting judgments of agents. The generality of AUT is derived from the logical classification of types (orders) of conflicts in the agent populations. At the first order of conflict, the two agent populations are disjoint and there is no interference of logic values assigned to any statement p and its negation by agents. The second order of conflict models superposition (interference) of logic values for overlapping agent populations where an agent assigns conflicting logic values (true, false) to the same attribute simultaneously.